Implicit sampling for an elliptic inverse problem in underground hydrodynamics

Implicit sampling is a Monte Carlo (MC) method that focuses the computational effort on the region of high probability by first locating this region via numerical optimization and then solving random algebraic equations to explore it. Implicit sampling has been shown to be efficient in online state estimation and filtering (data assimilation) problems; we use it here to estimate a diffusion coefficient in an elliptic equation, using sparse and noisy data. This problem has important applications in reservoir simulation/ management and in pollution modeling. We present an implementation of implicit sampling with a BFGS optimization coupled to an adjoint calculation, where random maps are used to solve the random algebraic equations. We perform numerical experiments to test the applicability and efficiency of our approach, and find that it can be more efficient than standard MC sampling by a large factor. We also show how to use multiple grids to further improve the computational efficiency of implicit sampling.